Georgia Institute of Technology College of Sciences

From a perspective of an applied algebraic geometer, I will address several use cases of algorithmic machinery that goes hand in hand with the language of tropical geometry. One of the examples originates in dynamical systems and may shed (tropical) light on a long-standing conjecture in celestial mechanics.

This talk will serve as an introduction to the algebra of hyperfields—fields with a multivalued addition. For example the sign hyperfield which is the arithmetic of real numbers modulo their absolute value (e.g. positive + positive = positive, positive + negative = any possibility). We will also introduce valued fields which capture the idea of how many times a fixed prime p divides the numerator or denominator of a rational number.

Using this arithmetic we will consider the combinatorial question of factoring a polynomial over a hyperfield. This will present a unified and conceptual way of looking at Descartes's rule of signs (how many positive roots does a real polynomial have) and the Newton polygon rule (how many roots are there which are divisible by p or p^2).

Tropicalization is usually applied to algebraic or semi-algebraic sets, but I would like to introduce a different category of sets with well-behaved tropicalization: sets with the Hadamard property, i.e. subsets of the positive orthant closed under coordinate-wise (Hadamard) multiplication. Tropicalization (in the sense of logarithmic limit sets) of a set S with the Hadamard property is a convex cone, whose defining inequalities correspond to pure binomial inequalities valid on S.

I will do several examples of sets S with the Hadamard property coming from combinatorics, such as counts of independent sets in matroids, counts of faces in simplical complexes, and counts of graph homomorphisms. In all of our examples we observe a fascinating polyherdrality phenomenon: even though the sets S we are dealing with are not semilagebraic (they are infinite subsets of the integer lattice) the tropicalization is a rational polyhedral cone. Also, the pure binomial inequalities valid on S are often combinatorially interesting.

Joint work with Annie Raymond, Rekha Thomas and Mohit Singh.

I will talk about how tropical geometry can be used for implicitization and elimination problems. Implicitization is the problem of finding the defining equations (implicit equations) of an algebraic variety from a given parameterization. Elimination is the problem of finding the defining equations of a projection of an algebraic variety. In some instances such as the case when the polynomials involved have generic coefficients, we give a combinatorial construction of the tropical varieties without actually computing the defining polynomials. Tropical varieties can then be used to compute invariants of the original varieties.